Non-Self-Adjoint Resolutions of the Identity and Associated Operators

Abstract

Closed operators in Hilbert space defined by a non-self-adjoint resolution of the identity \(\{X(\lambda )\}_{\lambda \in {\mathbb R}}\), whose adjoints constitute also a resolution of the identity, are studied. In particular, it is shown that a closed operator \(B\) has a spectral representation analogous to the familiar one for self-adjoint operators if and only if \(B=\textit{TAT}^{-1}\) where \(A\) is self-adjoint and \(T\) is a bounded inverse.